Since and , then .
Since and , then .
Answer: (D)
A square-based pyramid has 8 edges: 4 edges that form the square base and 1 edge that joins each of the four vertices of the square base to the top vertex.
Answer: (C)
Evaluating, .
Alternatively, we could notice that each of the numerator and denominator is a multiple of 20, and so .
Answer: (E)
The 7th oblong number is the number of dots in retangular grid of dots with 7 columns and 8 rows.
Thus, the 7th oblong number is .
Answer: (C)
Solution 1
Let the coordinates of be .
Since is the midpoint of and , then the difference between the -coordinates of and equals the difference between the -coordinates of and .
In other words, and so .
Similarly, and so .
Thus, the coordinates of are .
Solution 2
Let the coordinates of be .
The midpoint of and has coordinates .
Since is the midpoint of , then (which gives or ) and (which gives or ).
Therefore, the coordinates of are .
Answer: (B)
Each week, Carrie sends 5 messages to her brother on each of 2 days, for a total of 10 messages.
Each week, Carrie sends 2 messages to her brother on each of the remaining 5 days, for a total of 10 messages.
Therefore, Carrie sends messages per week.
In four weeks, Carrie sends messages.
Answer: (D)
Evaluating, .
Answer: (A)
Since , then .
Thus, and so .
Answer: (E)
Solution 1
Since of 60 is 12, then or .
Therefore, of is 15% of 20, or .
Solution 2
Since of 60 is 12, then or .
In terms of , 15% of equals or .
Since , then .
Answer: (D)
Solution 1
Since square has side length 2, then .
Since , , , are the midpoints of the sides of , then .
Since , then .
Therefore, square has side length .
The area of square is and the area of square is .
The ratio of these areas is or .
Solution 2
Join to and to .
Since is a square and , , , and are the midpoints of its sides, then and divide the square into four identical squares.
Each of these four squares is divided into two triangles of equal area by its diagonal. (These diagonals are , , , .)
Square is made up of 4 of these triangles of equal area.
Square is made up of 8 of these triangles of equal area.
Therefore, the ratio of these areas is or .
Answer: (A)
By the Pythagorean Theorem in , since .
Thus, .
By the Pythagorean Theorem in , since .
Therefore, the perimeter of is .
Answer: (B)
Since , its positive divisors are Of these, the integers are perfect squares, which means that 128 has three positive divisors that are perfect squares larger than 1.
Answer: (D)
Since form an arithmetic sequence, then the differences between consecutive terms are equal, or .
Thus, or and so .
Answer: (E)
Since the average of the four numbers is 13, then and so or .
Since and is an integer, then .
Since and , then is less than half of 26, or .
Since is an integer, then .
Therefore, we have .
There are 8 choices for in this range: 5, 6, 7, 8, 9, 10, 11, 12. (Note that .)
These give the pairs .
Thus, there are 8 possible pairs.
Answer: (B)
When Hicham runs 10 km at an average speed of 12 km/h, he takes hours to run this distance.
Since Hicham runs for a total of 1.5 hours, then he runs the last 6 km in hours.
Since he runs 6 km in hours, his average speed for this segment is km/h.
Answer: (B)
Solution 1
Since is a solution to the equation , then satisfies this equation.
Thus, and so or .
Therefore, the original equation becomes or .
Therefore, the other solution is .
Solution 2
We use the fact that the sum of the roots of an equation of the form is .
If the roots of the equation are and , then or .
Therefore, the other solution is .
Answer: (C)
The number of points on the circle equals the number of spaces between the points around the circle.
Moving from the point labelled 7 to the point labelled 35 requires moving points and so 28 spaces around the circle.
Since the points labelled 7 and 35 are diametrically opposite, then moving along the circle from 7 to 35 results in travelling halfway around the circle.
Since 28 spaces makes half of the circle, then spaces make the whole circle.
Thus, there are 56 points on the circle, and so .
Answer: (C)
The first equation gives and so or .
The second equation gives .
Multiplying this equation by 5 gives or .
Since , then or .
Rearranging, we obtain or .
Therefore, or .
Since , then or .
From the first equation, it cannot be the case that .
We can check that the pair satisfies both equations.
Therefore, .
Answer: (B)
Solution 1
Suppose that, when the students are put in groups of 2, there are complete groups and 1 incomplete group.
Since the students are being put in groups of 2, an incomplete group must have exactly 1 student in it.
Therefore, .
Since the number of complete groups of 2 is 5 more than the number of complete groups of 3, then there were complete groups of 3.
Since there was still an incomplete group, this incomplete group must have had exactly 1 or 2 students in it.
Therefore, or .
If and , then or and so .
In this case, and there were 15 complete groups of 2 and 10 complete groups of 3.
If and , then or and so .
In this case, and there were 14 complete groups of 2 and 9 complete groups of 3.
If , dividing the students into groups of 4 would give 7 complete groups of 4 and 1 incomplete group.
If , dividing the students into groups of 4 would give 7 complete groups of 4 and 1 incomplete group.
Since the difference between the number of complete groups of 3 and the number of complete groups of 4 is given to be 3, then it must be the case that .
In this case, ; the sum of the digits of is 12.
Solution 2
Since the students cannot be divided exactly into groups of 2, 3 or 4, then is not a multiple of 2, 3 or 4.
The first few integers larger than 1 that are not divisible by 2, 3 or 4 are 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35.
In each case, we determine the number of complete groups of each size:
|
5 |
7 |
11 |
13 |
17 |
19 |
23 |
25 |
29 |
31 |
35 |
# of complete groups of 2 |
2 |
3 |
5 |
6 |
8 |
9 |
11 |
12 |
14 |
15 |
17 |
# of complete groups of 3 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
# of complete groups of 4 |
1 |
1 |
2 |
3 |
4 |
4 |
5 |
6 |
7 |
7 |
8 |
Since the number of complete groups of 2 is 5 more than the number of complete groups of 3 which is 3 more than the number of complete groups of 4, then of these possibilities, works.
In this case, ; the sum of the digits of is 12.
(Since the problem is a multiple choice problem and we have found a value of that satisfies the given conditions and for which an answer is present, then this answer must be correct. Solution 1 shows why is the only value of that satisfies the given conditions.)
Answer: (B)
Since points , and form a straight line segment, then and so .
Since is the final position of after folding, then is the final position of after folding, and so .
Thus, and so .
Since and are parallel sides of the piece of paper, then , and so .
Finally, is opposite so .
Answer: (A)
When a marble is chosen from Box 1, the probability is that it will be gold and the probability is that it will be black.
Thus, after this choice is made, there is a probability of that Box 2 contains 2 gold marbles and 2 black marbles, and there is a probability of that Box 2 contains 1 gold marble and 3 black marbles.
In the first case (which occurs with probability ), the probability that a gold marble is chosen from Box 2 is and the probability that a black marble is chosen from Box 2 is .
In the second case (which occurs with probability ), the probability that a gold marble is chosen from Box 2 is and the probability that a black marble is chosen from Box 2 is .

Therefore, the probability that a gold marble is chosen from Box 2 is and the probability that a black marble is chosen from Box 2 is .
Thus, after this choice is made, there is a probability of that Box 3 contains 2 gold marbles and 3 black marbles, and a probability of that Box 3 contains 1 gold marble and 4 black marbles.
Finally, the probability that a gold marble is chosen from Box 3 equals the probability that Box 3 contains 2 gold marbles and 3 black marbles times the probability of choosing a gold marble in this situation (that is, ) plus the probability that Box 3 contains 1 gold marble and 4 black marbles times the probability of choosing a gold marble in this situation.
In other words, this probability is .
Answer: (A)
Solution 1
We expand the given expression to obtain We expand further to obtain We simplify to obtain We remove a common factor of 5 from the first two terms and then complete the square to obtain This gives or Since for all real numbers and for all real numbers , then the minimum value of (and hence of the original expression) is or 104.
We note that this value is actually achieved when (which gives ) and (which gives ).
Solution 2
We expand the given expression to obtain We expand further to obtain The terms involving are The terms involving are Since for all real numbers , then the minimum value of is 80.
Since for all real numbers , then the minimum value of is 24.
Since the minimum value of is 80 and the minimum value of is 24, then the minimum value of is .
Answer: (E)
We label the centres of the coins , as shown, and we join , , , , , , , , , , , and .

The radius of the circles with centres , and is 3 cm and the radius of the circles with centres , and is 2 cm. (From this point until the very last step of the problem, we will not include the units, which are always centimetres.)
Let the radius of the circle with centre be . We want to determine .
When we join the centres of two circles that are just touching, the resulting line segment passes through the point at which the circles touch and the length of this line segment is the sum of the radii of the two circles.
Thus, . Similarly, , , , and .
Now and each have side lengths 5, 5 and 6, which means that these triangles are congruent.
Similarly, and are congruent, and and are congruent.
Since corresponding angles in congruent triangles are equal, then is equal to .
But these six angles surround point , so the sum of these six angles is .
Thus, .
Now, we remove the circles from the diagram and focus on the top half of the diagram.
We join to , the midpoint of , and and to , the midpoint of .

Consider .
Since and is the midpoint of , then .
Also, since , then is isosceles and so is perpendicular to and bisects .
Similarly, is isosceles with and is isosceles with .
Since is the midpoint of and , then and and are perpendicular to at .
We know that .
But , so .
Also, , so .
Thus, .
Finally, we know that .
Since , then .
From this we obtain , then of the given choices, this radius is closest to .
(If more decimal places are carried through from earlier calculations, we obtain , which is still closest to .)
Answer: (D)
We begin by understanding and describing in a different way.
Each positive integer can be placed between two consecutive perfect squares. More precisely, for each positive integer , there exists a unique positive integer with .
Since , then and .
Since and are consecutive integers, then .
In other words, is the largest positive integer less than or equal to .
So we want to calculate the sum of all integers with for which is a multiple of its corresponding .
Now and .
This means that the possible values of with are the multiples of in this range, or , , and .
Since , then , and so .
So the given problem is equivalent to determining the sum of , and for each from 1 to 999.
Since , then we want to determine the sum of for to inclusive.
Thus, Since and for every positive integer , then and so .
Answer: (C)
We begin by partitioning the set into disjoint subsets of the form where is a positive integer with and is the largest non-negative integer with .
The first two of these sets are and .
These sets have and , and thus .
We will see that each element of is an element of exactly one of these sets.
Since , then for every positive integer , so is the largest possible value of that we can have.
For each with , we determine the range of values of that give that :
: since and
: since and
: since and
: since and
: since and
: since and
: since and
Since we want to create disjoint sets whose union is the set , then we exclude all ’s that are multiples of 3. (If were a multiple of 3, it would appear as an element of a set with .)
For each of the ranges above, we count the number of multiples of that we need to exclude:
Note for example that the range contains integers and includes the multiples of 3 from to , inclusive, so includes multiples of 3, and the range contains integers and includes the multiples of 3 from to , inclusive, so includes multiples of 3.
An integer in that is not a multiple of 3 generates a set and cannot appear in a different set . Any multiple of 3, say , will appear in exactly one where the value of (which is itself not a multiple of 3) is obtained by dividing out all of factors of 3 from .
We note that so the union of these sets includes enough elements to be the entire set . No element of appears in more than one , so the union of these sets is the entire set .
Why are these sets useful in the given problem?
A subset of is triple-free if it does not contain two elements, one of which is three times the other.
This means that a subset of is triple-free exactly when it does not contain two consecutive elements of any .
A triple-free subset of that contains as many elements as possible will contain as many elements as possible from each of the defined above.
Since no two consecutive elements of any can appear in , then a with 7 elements can contribute at most 4 elements to (every other element starting with the first), a with 6 elements can contribute at most 3 elements to (it cannot contain 4 elements without having two consecutive elements), and so on.
For each from to , each above can contribute at most the following number of elements to :
6 |
7 |
4 |
5 |
6 |
3 |
4 |
5 |
3 |
3 |
4 |
2 |
2 |
3 |
2 |
1 |
2 |
1 |
0 |
1 |
1 |
This means that can contain at most elements. This agrees with the information in the problem statement.
Where is there choice in creating ?
If a contains 1 element, then this element must be chosen for .
If a contains 3 elements, then there is 1 way to choose 2 elements for , since in this case the 1st and 3rd elements (in increasing order) must be chosen.
If a contains 5 elements, then there is 1 way to choose 3 elements for , since in this case the 1st, 3rd and 5th elements (in increasing order) must be chosen.
If a contains 7 elements, then there is 1 way to choose 4 elements for , since in this case the 1st, 3rd, 5th and 7th (in increasing order) elements must be chosen.
If a contains 2 elements, then there are 2 ways to choose 1 element for .
If a contains 4 elements, then there are 3 ways to choose 2 elements for . (If we want to choose 2 elements from without choosing consecutive elements, we can choose and , or and , or and .)
If a contains 6 elements, then there are 4 ways to choose 3 elements for . (If we want to choose 3 elements from without choosing consecutive elements, we can choose and and , or and and , or and and , or and and .)
This means that the number of ways of choosing a triple-free subset of with as many elements as possible is .
This is because there are 2 sets with and 1 way of choosing 4 elements from each of them (giving choices in total), 4 set with and 4 ways of choosing 3 elements from each of them (giving choices in total), and so on.
The seven factors in this product give, from to , the number of ways of choosing the maximum number of elements from a set corresponding to that value of raised to the power of the number of such sets .
Therefore, the number of ways equals .
From the statement of the problem, we thus have .
The final three digits of this integer are .
Answer: (A)